Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 43.2 months and a standard deviation of 9.3 months.
$\Omega$ USE SALT
(a) If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of its batteries will the company expect to replace? (Round your answer to two decimal places.)
$\%$
(b) If Quick Start does not want to make refunds for more than $15 \%$ of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?
months

Step 1 ：The problem is asking for two things. First, it wants to know the percentage of batteries that the company will expect to replace if it guarantees a full refund on any battery that fails within the 36-month period after purchase. This is a question about the cumulative distribution function (CDF) of a normal distribution. The CDF at a point x is the probability that a random variable drawn from the distribution will be less than or equal to x. In this case, we want to find the CDF at 36 months, which will give us the percentage of batteries that fail within this time.

Step 2 ：Second, the problem asks for how long the company should guarantee the batteries if it does not want to make refunds for more than 15% of its batteries. This is a question about the inverse cumulative distribution function (also known as the quantile function) of a normal distribution. The quantile function at a probability p gives the value x such that the CDF at x is equal to p. In this case, we want to find the quantile at 15%, which will give us the number of months for which the company should guarantee the batteries.

Step 3 ：The mean life of a Quick Start battery is 43.2 months and the standard deviation is 9.3 months.

Step 4 ：The cumulative distribution function at 36 months is approximately 0.2194, or 21.94% when expressed as a percentage. This is the percentage of batteries that the company will expect to replace if it guarantees a full refund on any battery that fails within the 36-month period after purchase.

Step 5 ：The inverse cumulative distribution function at 15% is approximately 33.56, which we can round to 34 months to the nearest month. This is the number of months for which the company should guarantee the batteries if it does not want to make refunds for more than 15% of its batteries.

Step 6 ：Final Answer: (a) The company will expect to replace approximately \(\boxed{21.94\%}\) of its batteries. (b) The company should guarantee the batteries for \(\boxed{34}\) months.